## Table of Contents:-

- What is Hypothesis?
- Formulation of Hypothesis
- Hypothesis Testing
- Element of Hypothesis Testing
- Advantages of the Tests of Hypothesis
- Disadvantages of the Tests of Hypothesis

## What is Hypothesis?

The term hypothesis is derived from the Greek word “hypotithenai,” which means “to put under” or “to suppose.”

A hypothesis is the process of using sample statistics to make inferences for a population parameter.

**According to Prof. Morris Hamburg,** “A hypothesis in statistics is simply a quantitative statement about a population.”

**According to Palmer O Johnson,** “A hypothesis as islands in the uncharted seas of thought to be used as bases for consolidation and recuperation as we advance into the unknown.”

A hypothesis generally arises when we engage in inductive reasoning. In this process, the researcher gathers a series of observations to develop a theory. Thus, a hypothesis serves as the initial step in any research endeavour. It is essential in research and can be formulated in various ways. Its fundamental purpose lies in predicting the final result of the investigation.

## Formulation of Hypothesis

**Step 1) Establishing a Hypothesis**

To formulate a hypothesis, the first step is to define the null hypothesis (denoted by H_{0}_{.}). The null hypothesis is assessed for potential rejection, operating under the assumption that it is true. It is generally expressed in equation form.

The null hypothesis is given as follows:

H_{0 }: Î¼ = Î¼_{0}

Where,Â

- Î¼ = Population Mean andÂ
- Î¼
_{0 = }Hypothesized value of the Population Mean.

**For example,** to assess whether a population mean equals 160, the null hypothesis can be framed as “the population mean is equal to 160.”

H_{0}: Î¼ = 160Â

Now, establish the alternative hypothesis, denoted as H_{1}, which is logically opposite to the null hypothesis.

**According to Prof R.A. Fisher,** “Null hypothesis is the hypothesis which is tested for possible rejection under the assumption that it is true.”

* Note: *When the null hypothesis is true, the alternative hypothesis must be false, and vice versa.

Alternative hypothesis is given as:

H_{0 }: Î¼ â‰ Î¼_{0}

Therefore, H_{0 }: Î¼ < Î¼_{0}

H_{0 }: Î¼ > Î¼_{0}

For the example provided above, the alternative hypothesis can be formulated as “the population mean is not equal to 160.”

so, H_{1}_{Â }: Î¼ â‰ 160

This provides two other alternative hypotheses:

H_{1}: Î¼ < 160, Indicates i.e., the population mean is less than 160 and

H_{1}: Î¼ > 160 Indicates i.e., the population mean is greater than 160.

**Step 2) Establish a Suitable Significance Level**

The significance level, denoted by Î±, is linked to the null hypothesis. It represents the size of the rejection or critical region.

Ensure that the obtained result is free from the decision-maker’s bias of choice. To achieve this, the significance level must always be determined before drawing samples.

0.01, 0.05, and 0.10 are the significance levels most commonly used by researchers.

**Step 3) Test Statistic**

In this step, one can select the suitable statistical test for the analysis. The choice of statistical test depends on the type, number, and level of data. To select the appropriate statistical test, the statistics (mean, proportion, etc.) used in the study are also considered to achieve accurate results.

**Step 4) Performing Computations**

After selecting the statistical test, the researcher tests the performance of various calculations using a random sample of sizes. Here, the researcher also computes the standard error.

** Step 5) Making a Decision**

In this step, the researcher decides whether to reject or accept the null hypothesis.

The following conditions are checked to determine whether to accept or reject a null hypothesis.

**1)** When the calculated value is less than the critical value, the test statistic falls within the acceptance region, and the null hypothesis is accepted.

**2)** When the calculated value is greater than the critical value, the test statistic falls within the rejection region, and the null hypothesis is rejected.

Generally, in hypothesis testing, a significance level of 5% (Î±=0.05) is commonly used to make the decision.

## Hypothesis Testing

Hypothesis testing plays an essential role in applying statistics to real-life problems. It helps researchers make decisions by using sampled data from an unknown population distribution and its parameters.

For example, consider a manufacturing company that produces geysers. The average life of the geysers manufactured using the old process is 1000 hours, while the average life is 1500 hours for those manufactured using the new method.

So, in this case, we can formulate three hypotheses:

- The new process is better than the old process.
- The old process is better than the new process.
- There is no difference between both processes.

### Hypothesis Decision Table

These tables are combinations of rows and columns, each corresponding to a single rule. The columns in the table describe the actions and conditions for the rules. When every condition is valid, the action of a given rule is performed. The table below displays a hypothesis decision table.

Hypothesis Decision Table |
||

Decision | Null hypothesis | |

AcceptÂ | Correct Decision | Type II Error |

RejectÂ | Type I Error | Correct DecisionÂ |

## Element of Hypothesis Testing

Given below are the two basic elements of hypothesis testing.

### 1) Test Statistic

The test statistic is determined from the sample. Researchers test the hypothesis by assuming that the population parameter is true, and then they compare the actual value with the hypothetical value of the parameter. If the difference between the values is less, the hypothesis is accepted; otherwise, it is rejected.

A statistic upon which the decision to accept or reject a hypothesis can be based is called a Test Statistic (T.S.). Here are examples of test statistics:

$$\style{font-family:Arial}{\style{font-size:16px}{\mathrm T.\mathrm S.\;\;\mathrm Z\;=\;\frac{({\overline{\mathrm X}}_1-{\overline{\mathrm X}}_2)}{\sqrt{\left(\frac{{\mathrm S}_1^2}{{\mathrm n}_1}+\frac{{\mathrm S}_2^2}{{\mathrm n}_2}\right)}}\;(\mathrm{For}\;\mathrm Z-\mathrm{test},\;\mathrm{for}\;\mathrm{two}\;\mathrm{samples})}}$$

In the case of a single sample, the test statistic (T.S.) is given as follows:

$$\style{font-family:Arial}{\style{font-size:16px}{\mathrm T.\mathrm S.\;\;\mathrm Z\;=\;\frac{\overline{\mathrm X}-\mathrm\mu}{\displaystyle\frac{\mathrm\sigma}{\sqrt{\mathrm n}}}\;\mathrm{where},\;\frac{\mathrm\sigma}{\sqrt{\mathrm n}}}}$$

### 2) Rejection Region

The rejection region is a subset of the sample space and is closely linked to the statistical test being conducted. If the observed sample falls only within this set, then the researcher can reject the null hypothesis and choose its alternative.

**Example:** A survey was conducted where 50 workers were questioned about the number of hours they slept each day. One wants to test the hypothesis that smokers need less sleep than the general public which needs an average of 7.7 hours of sleep. The steps below are as follows:

** i)** Calculate a rejection region for a significance level of 0.05.

**ii)** If the sample mean is 7.5 and the population standard deviation is 0.5, what conclusion can be made?

**Solution:**The null and alternative hypotheses are given as follows:

- H
_{0}: Î¼ = 7.7 - Î—
_{1Â }= Î¼ < 7.7

This is a left-tailed test. The z-score corresponding to a significance level of 0.05 is -1.645. The critical region is defined as the area that lies to the left of -1.645. If the z-value is less than -1.645, then one will reject the null hypothesis and accept the alternative hypothesis. If the test statistic is greater than -1.645, then one will fail to reject the null hypothesis and conclude that the test was not statistically significant.

$$\style{font-family:Arial}{\style{font-size:16px}{Z=\frac{7.5-7.7}{0.5/\sqrt{50}}=\;-2.83}}$$

As -2.83 is to the left of -1.645, it falls within the critical region. Therefore, the null hypothesis is rejected in favour of the alternative hypothesis. One can conclude that smokers need less sleep.

## Advantages of the Tests of Hypothesis

**1)** It is suitable for establishing the focus and direction of a research effort.

**2)** It helps the researcher in developing a hypothesis, which in turn shapes the purpose of the research effort.

**3)** It determines which variables will not be measured in a study and similarly identifies those that will be measured.

## Disadvantages of the Tests of Hypothesis

**1) **Results of significance tests are based on probabilities and, therefore, cannot be expressed with absolute certainty. When a test shows that a difference is statistically significant,Â it implies that the difference is likely not attributable to random chance.

**2) **Statistical inferences based on significance tests should not be considered definitive evidence regarding the validity of hypotheses. This is especially so in the case of small samples where the probability of drawing erring inferences happens to be generally higher. For greater reliability, the size of the samples is sufficiently enlarged.

**3) **Tests do not explain the reasons why the difference exists, say between the means of the two samples. They only show if the difference is caused by sampling fluctuations or other reasons. However, the tests do not provide information about the specific reasons behind the difference.

**4) **The tests should not be used mechanically. It should be kept in view that testing is not decision making itself, the tests are only useful aids for decision-making. Therefore, it is essential to interpret statistical evidence correctly to make informed decisions.